Measurement of third-order elastic constants and applications to loaded structural materials

Static stress
Tij

Following Murnaghan’s theory, static stress
Tij

is written as

Tij=Ji??(?0?)???j(7)

where
Ji?

is the Jacobian matrix,
?0? the free energy per unit volume of deformed isotropic solid, and
?0 the density of isotropic solid in non-deformed state. The free energy per unit volume
can be written in terms of the strain invariants
I1,I2,I3

?0?=A0I1+(?+2?)2I12-2?I2+(?+m)3I13-2mI1I2+nI3(8)

and hence its derivatives with respect to the static strain coefficients are

?(?0?)??ij=(?+2?)I1+(?+2m)I12-2mI2?I1??ij-(?+mI1)?I2??ij+n?I3?ij(9)

where
A0=0,

? and
? are the Lamé constants,
l,m,n are Murnaghan’s third-order elastic constants, and the strain invariants are defined
by

I1=?11+?22+?33=??????

I2=?22?23?32?33+?33?31?13?11+?11?12?21?22=12(??????-??????)(10)

I3=?11?12?13?21?22?23?31?32?33=12???(??????-??????),(?????)(11)

The derivatives of the above invariants of
I1,I2

and
I3 are given as,

?I1??11=1,?I1??23=0,?I2??11=?22+?33=I1-?11,?I2??23=-?32,?I3??11=?22?33-?23?32,(12)

?I3??23=?12?31-?11?32=-?11?12?31?32(13)

Thus the derivative of the free energy
?0?

with respect to
?11, for example, becomes

?(?0?)??11=(?+2?)I1+(?+2m)I12-2mI2-2(?+mI1)(I1-?11)+n(?22?33-?23?32)(14)

See Appendix B on the derivative with respect to
?22

,
?33,
?23,
?31,
?12.

Infinitesimal stress of elastic wave
T^ij

The total stress
T¯ij

is defined as the sum of the infinitesimal stress
T^ij and the static stress
Tij, that is,

T¯ij=(Tij+T^ij)(15)

Using formula (7) and replacing
?

for
?0?, formula (15) is rewritten as

T¯ij=J¯i?(??¯???j)=(Ji?+J^i?)(?????j+??^???j)=Ji??????j+J^i??????j+Ji???^???j+J^i???^???j(16)

T^ij=J^i??????j+(Ji?+J^i?)??^???j(17)

Here the elements of the Jacobian matrix are expressed as

J¯i?=?xi?ai,Ji?=?Xi?a?=?(ai+Ui)?a?=?i?+?Ui?a?,J^i?=J¯i?-Ji?=?xi?a?-?Xi?a?=?ui?a?(18)

Using the above formula (18),

J11=?X1?a1=1+?U1?a1,J23=?X2?a3=?U2?a3,J^11=?x1?a1-?X1?a1=?u1?a1,J^22=?u2?a2,J^33=?u3?a3(19)

J^23=?x2?a3-?X2?a3=?u2?a3,J^31=?u3?a1,J^12=?u1?a2(20)

See Appendix C about expressions for the derivatives of
?u1/?a1

,
?u1/?a2, and
?u1/?a3.

T^ij?J^i??????j+Ji???^???j(21)

Setting
?=1

, and
j=1 in the
??^/???j, yield

??^??11=??¯??11-????11(22)

Expanding formula (22) using formula (14) then gives

??^??11=?I¯1+?I¯12-2mI¯2+2?¯11(?+mI¯1)+n(?¯22?¯33-?¯23?¯32)-?I1-?I12+2mI2-2?11(?+mI1)-n(?22?33-?23?32)?(?+2?+2?I1+4m?11)?^11+(?+2?I1-(2m-n)?33)?^22+(?+2?I1-(2m-n)?22)?^33+(2m-n)(?23?^32+?32?^23)+2m(?13?^31+?31?^13+?12?^21+?21?^12)(23)

See Appendix D for a detailed derivation. Other derivatives are similarly obtained

An expression for
??^

/
??22 can be obtained through changing subscripts 1
?2, 2
?3, 3
?1 in the formula 23).

The formula of
??^/??33

can be obtained in a similar manner. Also

??^??12?(2m-n)?21?^33+2m?21(?^11+?^22)+(2?+2mI1-n?33)?^21+n(?31?^23+?23?^31)(24)

and hence by
1?2

??^??21?(2m-n)?12?^33+2m?12(?^22+?^11)+(2?+2mI1-n?33)?^12+n(?32?^13+?13?^32)(25)

Using the above formulas (21) to (25), formula (17) for the infinitesimal stress is rewritten as follows.

T^11=J^1??????1+J1???^???1??+2?+(?+2?)I1+(3?+8?+4m)?1?1?+?+(?+2?)I1+??2-(?+2m-n)?3?2?+?+(?+2?)I1+??3-(?+2m-n)?2?3?+12(?+2m-n)?4?4?+12(2?+3?+2m)(?5?5?+?6?6?)+2??12?u1?X2+?13?u1?X3(26)

See Appendix E for details of the derivation for
T^11

. Also,

T^22?(?+2?+(?+2?)I1+(3?+8?+4m)?2?2?+?+(?+2?)I1+??3-(?+2m-n)?1?3?+?+(?+2?)I1+??1-(?+2m-n)?3?1?+12(?+2m-n)?5?5?+12(2?+2m+3?)(?4?4?+?6?6?)+2??23?u2?X3+?21?u2?X1(27)

T^33??+2?+(?+2?)I1+(3?+8?+4m)?3?3?+?+(?+2?)I1+??1-(?+2m-n)?2?1?+?+(?+2?)I1-(?+2m-n)?1+??2?2?+12(?+2m-n)?6?6?+12(2?+2m+3?)(?4?4?+?5?5?)+2??31?u3?X1+?32?u3?X2(28)

In the expression for
T^ij

, the subscripts of 1, 2, and 3 for
?? and
??´ change cyclically with 1
? 2, 2
? 3, 3
? 1. Similarly, indices 4, 5, 6 change cyclically like 4
? 5, 5
? 6, 6
? 4.

For
???

and
?u?/?X?, the subscripts change cyclically as well with 1
? 2, 2
? 3, 3
? 1. Hence from

T^23=J^2??????3+J2???^???3=2??31?u2?X1+(?I1+2??33)?u2?X3+12((?+2m-n)?1?+12(?+4?+2m)?2?+12(?+4?+2m)?3?)?4+122?+2(?+m)I1-(2?+n)?1+2??2?4?+14(4?+n)?6?5?+14(2?+n)?5?6?(29)

we obtain

T^32=(?I1+2??22)?u3?X2+2??21?u3?X1+12((?+2m-n)?1?+12(?+2m+4?)?2?+12(?+4?+2m)?3?)?4+12(2?+2(?+m)I1-(2?+n)?1+2??3)?4?+14(4?+n)?5?6?+14(2?+n)?6?5?(30)

See 13 about induction process of
T^11

.

T^31=(?I1+2??11)?u3?X1+2??12?u3?X2+12(?+2?+2m-n)?2?+12(?+4?+2m)?3?+12(?+4?+2m-n)?1?+12(2?+2(?+m)I1-(2?+n)?2+2??3)?5?+14(4?+n)?4?6?+14(2?+n)?6?4?(31)

T^13

can be obtained from formula (31) for
T^31 by substituting elements as follows:

?u3/?X1??u1/?X3,?u3/?X2??u1/?X2,?4?6´??6?4´,?6?4´??4?6´

and retaining the elements within the parentheses as these are unaffected by the interchange
in the formula of
T^13

.

T^12

is expressible as

T^12=(?I1+2??22)?u1?X2+2??23?u1?X3+12(?+2?+2m-n)?3?+12(?+4?+2m)?1?+12(?+4?+2m-n)?2?+12(2?+2(?+m)I1-(2?+n)?3+2??1)?6?+14(4?+n)?5?4?+14(2?+n)?4?5?(32)

from which
T21^

can be obtained by changing elements
?u1/?X2??u2/?X1,
?u1/?X3??u2/?X3,
?5?4´??4?5´,
?4?5´??5?4´, while retaining the elements in the parentheses as these are unaffected by the index
interchanges.

Propagation velocity of elastic wave to the direction of static uniaxial stress

The infinitesimal displacement of an elastic wave
ui

is expressed as

ui=Aexpi(?t-?Xi)(33)

where A is the amplitude,
?

the angular frequency,
? the wave number, and
i the imaginary unit.

The equation of motion for an elastic wave is written as

?0?2ui?t2=?T^i??a?=?T^i??X??X??a?(34)

The various expansions of formula (34) are given as (A) to (E) as bellow:

(A) For longitudinal wave

?0?2u1?t2=?T^1??X??X??a?=T^11?X??X??a1+T^12?X??X??a2+T^13?X??X??a3(35)

?Ui/?aj=0

,
(i?j) The expansion of the above formula (35) is

?0?2u1?t2=?T^11?X11+?U1?a1+?T^12?X21+?U2?a2+?T^13?X31+?U3?a3(36)

u1=Aexpi(?t-?X1),u2=u3=0?2?=?3?=?4?=?5?=?6?=0

Accordingly,

?0?2u1?t2=(1+?1)??X1(?+2?+(?+2?)I1+(3?+8?+4m)?1)?1?(37)

From the expressions for
?2u1/?t2

,
?/?X1, and
?1? of the above formula (37),

?2u1?t2/??1??X1=(?/k)2=V112(38)

and hence

?0V112=(1+?1)(?+2?+(?+2?)I1+(3?+8?+4m)?1)??+2?+T11E(?+2?+?+2?+3?+8?+4m-2?(?+2?))=?+2?+T11E(5?+10?+2?+4m-2?(?+2?))(39)

where the term quadratic in strain,
?12

is neglected, and we have used

?1=?U1?a1=T11E,?2?1=?3?1=-?(40)

where
?

is Poisson’s ratio, and E is Young’s modulus

(B) For transverse wave

?0?2u2?t2=1+?U1?a1?T^21?X1+1+?U2?a2?T^22?X2+1+?U3?a3?T^23?X3(41)

u2=Aexpi(?t-kX1),u1=u3=0,?6??0

the others are 0,

?Ui?aj=0,i?j

then

?0?2u2?t2=(1+?1)((?I1+2??11)+12(2?+2(?+m)I1-(2?+n)?3+2??2))?2u2?X12(42)

?2u2?t2/?2u2?X12=(?/k)2=V122(43)

?0V122=(1+?1)(?+(?+?+m)I1+2??11+??2-12(2?+n)?3)=?+T11E(?+?+?+m+2?-?2?+2?+2m+?-?-n2)=?+T11E(?+4?+m-?2?+2?+2m-n2)(44)

where the term of
?12

is also neglected in a similar way to formula (39).

(C) For transverse wave

?0?2u1?t2=1+?U1?a1?T^11?X1+1+?U2?a2?T^12?X2+1+?U3?a3?T^13?X3u1=Aexpi(?t-kX2),u2=u3=0(45)

?0?2u1?t2=(1+?2)(?I1+2??2+?+(?+m)I1-?+n2?3+??1)?u1?X3(46)

?0V212=(1+?2)?+(?+?+m)I1+??1+2??2-?+n2?3=?+T11E?+?+m-?2?+4?+2m-n2(47)

(D) For longitudinal wave

?0?2u2?t2=(1+?U1?a1)?T^21?X1+(1+?U2?a2)?T^22?X2+(1+?U3?a3)?T^33?X3u2=Aexpi(?t-kX2),u1=u3=0(48)

?0?2u2?t2=(1+?2)(?+2?+(?+2?)I1+(3?+8?+4m)?2)?2u2?X22(49)

?0V222=?+2?+T11E?+2?-?(?+2?+2?+4?+3?+8?+4m)=?+2?+T11E?+2?-?(6?+10?+4?+4m)(50)

(E) For transverse wave

?0?2u3?t2=1+?U1?a1?T^31?X1+1+?U2?a2?T^32?X2+1+?U3?a3?T^33?X3u3=Aexpi(?t-kX2),u1=u2=0(51)

?0?2u3?t2=(1+?2)(?I1+2??2+?+(?+m)I1-?+n2?1+??3)?2u3?X22(52)

?0V232=(1+?2)?+(?+?+m)I1-?+n2?1+2??2+??3=?+T11E?+?+m-?-n2-?(?+2?+2?+2m+2?+?)=?+T11E?+m-n2-?(2?+6?+2m)(53)